By David F. Walnut
This booklet presents a finished presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the significant principles of wavelet idea via delivering a close exposition of the Haar sequence, then exhibits how a extra summary procedure permits readers to generalize and increase upon the Haar sequence. It then provides a few diversifications and extensions of Haar development.
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Additional resources for An Introduction to Wavelet Analysis
1/72?. However, 011 average, f, (z) is close t o the lirnit function. 25(c) tells a different story. The sequence converges pointwise but not uniformly to f (z) Y 0 on [O. 11, but since the area under the graph of f,, (x) is always 1, f,(z) does not converge to f ( z ) in mean. The width of the triangle under the graph of f (z) decreases t o zero, but the height increases to infinity in such a way that the area of the triangle does not go t o zero. The above examples show that sometimes pointwise convergence and mean convergence go together and sometimes they do not.
12. 24. 13. 25. 26. 10. 2 a(n) if n is even, if n is odd. = a , then liin,,, a(n) = a, Approximate Identities The notion of an approximate identity or summability kernel is used extensively in all branches of analysis. ionn that is well known and widely used by physicists. engineers, and mathematicians. The delta function, 6(z),has the property that for any continuous function f (x), or more generally, f (t) d(x - t) d t = f (4 for. every z E R. From sorrle elernenlary considerations he reader.
27. For each lc E N , and a > 0, define the Dirichlet kernel Uk (x) b y k D&) = C ,2nimx/a. 5. good exposiliorls uf this theory can be found in Horvath, A n introduction t o distributions, The American Mathematical Monthly, vol. 77 (1970) 227-240, and Benedetto, Harmonic Analysis and Applications, CRC Press (1997). 2. 28. 38) and reqi~iresonly t,he formula for summing a geometric series. ve periocl a. 29. For each n E N , and a > 0, define the Fejhr kernel F,(x) bY n- 1 CD~(X). 6. 30. For each n E N , and a > 0 , the Fej6r kernel, F,(x),can he written as and for any period a function f ( x ) , :La a ( n )( x ) = - f (X - t ) F,(t) d t .
An Introduction to Wavelet Analysis by David F. Walnut